1. What an Empty Relation Means
An empty relation contains no ordered pairs at all. Even though the sets may have many elements, the relation chooses none of them. So the relation is simply the empty set.
2. Formal Definition
If A is a set, then the empty relation on A is:
R = \emptyset
This means no element of A is related to any element of A (or any element of another set).
3. Why It Is Called “Empty”
The Cartesian product may have many ordered pairs, but the empty relation selects none of them. So the relation exists, but it has zero connections.
3.1. Illustration
A = \{1,2,3\}
A \times A = \{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}
Empty relation: R = \emptyset
4. Properties of the Empty Relation
Even though R has no pairs, it still satisfies some properties automatically.
4.1. Observations
- Not reflexive (because no (a,a) is included).
- Irreflexive (because no self-pairs appear).
- Symmetric (nothing violates symmetry).
- Antisymmetric (no pair breaks the rule).
- Transitive (no chain of pairs can break transitivity).
5. Example for Understanding
If A = \{x,y,z\}, then:
R = \emptyset
No element is connected to any other element. This represents complete “no relation”.